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The Hill sphere estimates the region around a small secondary where its gravity competes successfully with the primary's tidal field. For secondary mass $m\ll M$ on a nearly circular orbit of radius $a$, its characteristic Hill radius isIn the local rotating approximation, balancing $Gm/r^2$ against the radial tidal-plus-centrifugal term $3GMr/a^3$ gives this scale. It also approximates the distance to the adjacent Collinear Lagrange points.

This is a scale for orbital dynamics, not a surface where the two gravitational attractions become equal. Satellite stability depends on the trajectory and orbital orientation; being inside the nominal Hill sphere alone does not prove long-term stability. The small-mass-ratio collinear Lagrange points make the underlying approximation explicit.

A tidal field describes the spatial variation of gravitational acceleration across an extended body. With $g_i=-{\partial }_i\Phi$, a small separation $\xi$ gives relative acceleration $\delta g_i=T_{ij}{\xi }_j$, where $T_{ij}=-{\partial }_i{\partial }_j\Phi$.

For a point mass, a radial separation is stretched at rate coefficient $2GM/r^3$, while transverse separations are compressed with coefficient $-GM/r^3$. The common acceleration can be removed by choosing a freely falling frame; these relative accelerations remain. This makes tides a probe of gravitational inhomogeneity rather than merely the strength of the local acceleration.